Xiaofeng Wang

Purpose: The purpose of this paper is to study local and semilocal convergences of a fourth-order derivative-free iterative method in Banach spaces and its applications. Design/methodology/approach: The local convergence and semilocal convergence of a fourth-order derivative-free iterative method are analyzed under the assumptions that the first-order divided differences satisfy the weak Lipschitz continuity conditions. Findings: In local convergence, the domain of convergence and the error estimation can be obtained. In semilocal convergence, convergence criteria can be obtained, based on any initial point, to ensure that the iterative sequence converges to a unique solution in a given domain. Both of the above convergence analyses can prove the uniqueness of the solution. Originality/value: In the proof of convergence order, Taylor expansions require third or higher derivatives. The applicability of the method is restricted. To extend the applicability of the method, local and semilocal convergences are studied under Lipschitz conditions with the first derivative. Finally, to prove the applicability of the method, the method is applied to solve ordinary differential equations, partial differential equations and Beidou satellite positioning. The comparison between the method and the same order methods shows that the method has good behavior of convergence.

Based on the Lipschitz continuity condition, we study the local convergence analysis of a fifth-order Newton-type iterative method to solve nonlinear equations in Banach spaces. Lipschitz continuity condition on the first derivative is assumed to broaden the applicability of the method. Our study provides a thorough local convergence analysis in Banach spaces, including radii of convergence balls, computable error bounds, and the uniqueness of the solution. Finally, numerical experiments on various nonlinear models confirm the theoretical convergence criteria.

First, a Kurchatov-type derivative-free single-parameter method without memory for solving nonlinear systems is constructed, and the convergence order of the method is proved to be third order by using the higher-order Fréchet derivative. By using the Kurchatov-type difference operator to represent the variable parameter, the method without memory develops into two Kurchatov-type methods with memory, and the convergence order of the two methods with memory can reach 3.303 and 3.414, respectively. The new Kurchatov-type methods with and without memory use LU decomposition only once per iteration and require less computational cost when applying them to solve nonlinear systems. Second, in order to enlarge the application range of the novel Kurchatov-type method, the local convergence of the new Kurchatov-type method with and without memory is proved by using the [ ]-continuity condition of the first Fréchet derivative. In this way, the influence of higher-order derivatives on the proof of convergence is avoided, and the application range of the method is expanded. In addition, the radius of convergence around the solution is obtained by theoretical proof, and the uniqueness of the solution is proved. Finally, convergence radius is calculated by solving nonlinear systems with Kurchatov-type methods in numerical experiments, and the error and calculation time of different iteration methods are compared.

Purpose: The purpose of this paper is to study local convergence and applications of a new seventh-order iterative method.Design/methodology/approach: The order of convergence for the method is proved by using Taylor expansions. In addition, local convergence is studied under Lipschitz conditions with the first derivative.Findings: By using Taylor expansions, we can show the convergence order of the method is seven. The specific domains of convergence and the solutions of nonlinear equations can be obtained by applying the method to practical physics problems and nonlinear systems. In this way, the uniqueness of the solution and error estimates also are analyzed.Originality/value: In the proof of convergence order, Taylor expansions require third or higher derivatives. The applicability of the method is restricted. In order to extend the applicability of the method, local convergence is studied under Lipschitz conditions with the first derivative. Finally, in order to prove the applicability of the method, the method is applied to some physical problems and nonlinear systems.

In this paper, the local convergence of a high-order Chebyshev-type method without the second derivative is studied. We study the convergence under $ \omega $-continuity conditions based on the first derivative. The uniqueness of the solution and the radii of convergence domains are obtained. In contrast to the conditions used in previous studies, the new conditions of convergence are weaker. In addition, the attractive basins of the family with different parameters are studied, which can show the different stability of the family. Finally, in numerical experiments, the iterative method is used to solve different nonlinear models, including vertical stresses, civil engineering problem, blood rheology model, and so on. Theoretical results of convergence criteria are verified.